Problem: Solve for $x$ : $ 3|x - 4| - 6 = -2|x - 4| + 5 $
Explanation: Add $ {2|x - 4|} $ to both sides: $ \begin{eqnarray} 3|x - 4| - 6 &=& -2|x - 4| + 5 \\ \\ { + 2|x - 4|} && { + 2|x - 4|} \\ \\ 5|x - 4| - 6 &=& 5 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 5|x - 4| - 6 &=& 5 \\ \\ { + 6} &=& { + 6} \\ \\ 5|x - 4| &=& 11 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x - 4|} {{5}} = \dfrac{11} {{5}} $ Simplify: $ |x - 4| = \dfrac{11}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 4 = -\dfrac{11}{5} $ or $ x - 4 = \dfrac{11}{5} $ Solve for the solution where $x - 4$ is negative: $ x - 4 = -\dfrac{11}{5} $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& -\dfrac{11}{5} \\ \\ {+ 4} && {+ 4} \\ \\ x &=& -\dfrac{11}{5} + 4 \end{eqnarray} $ Change the ${ + 4}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{11}{5} {+ \dfrac{20}{5}} $ $ x = \dfrac{9}{5} $ Then calculate the solution where $x - 4$ is positive: $ x - 4 = \dfrac{11}{5} $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& \dfrac{11}{5} \\ \\ {+ 4} && {+ 4} \\ \\ x &=& \dfrac{11}{5} + 4 \end{eqnarray} $ Change the ${ + 4}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{11}{5} {+ \dfrac{20}{5}} $ $ x = \dfrac{31}{5} $ Thus, the correct answer is $x = \dfrac{9}{5} $ or $x = \dfrac{31}{5} $.